Comments...

A simple DC motor model is shown below. It consists of a rotating mass, an input torque and viscous friction.

where J=10and b=20

The input is torque (T) and the output is motor position (θ)

The system transfer can be found as follows:

J

J

substituting in value for b and J gives d^{2}θ_________dt^{2}

= ∑ Torque J

d^{2}θ_________dt^{2}

= T - b d θ_________dt

d^{2}θ_________dt^{2}

+b_________J

dθ_________dt

= 1_________J

T d^{2}θ_________dt^{2}

+2 dθ_________dt

= 0.1 T T_________θ

= 0.10_________s^{2} + 2 s

We wish to control the motor position (θ) and have a damping ratio (ζ) of 0.707

The open characteristics of the system, shown below, don't meet the design specification. First the system does not have a ζ of 0.707. Second, the output is an integral of the input (pole at the origin). We wanted the output to follow the input.

Fig 6.1 Open-Loop Analysis 0:00

T = k_{p}(θ_{desired} - θ)

Substituting into the differential equation for the system gives

d^{2}θ_________dt^{2}

+2 dθ_________dt

= 0.1 kd^{2}θ_________dt^{2}

+2 dθ_________dt

+ 0.1 kThis can also be done using block diagrams and transfer functions

where θ is the measured position, θNow simplify the transfer function

G(s)_________1+G(s)

= 0.1 k_{p}_________s^{2} + 2 s

1+

0.1 k_{p}_________s^{2} + 2 s

0.1 k_{p}_________s^{2} + 2 s + 0.1 k_{p}

Note that when analyzing the closed loop system using root locus the characteristic equation is

s^{2} + 2 s + 0.1 k_{p}

which d(s) = s^{2} + 2 s and n(s) = 0.1

which d(s) = s

We can find k_{p} using either root locus or, if we know the desired closed-loop poles, we can find k_{p} directly through algebra.

Given the closed loop characteristic equation

s^{2} + 2 s + 0.1 k_{p} = 0

where d(s) = s^{2} + 2 s and n(s) = 0.1

plot the poles (roots of d(s)) and zeros (roots of n(s)) as 0 ≤ kwhere d(s) = s

From all of the possible solutions choose k_{p} that results in ζ=0.707. Do this using brute force by selecting a k_{p} and finding the roots of the resulting quadratic equation. As you change k_{p} the root locations are indicated with an x on the root locus plot.

s^{2} + 2 s + 0.1 k_{p} = 0 where k_{p} =

roots are: s = 0, -2

0.000

roots are: s = 0, -2

If you know the form of the characteristic equation for the closed-loop system, you can often solve directly for k_{p}. In our problem closed-loop characteristic equation is:

s^{2} + 2 s + 0.1 k_{p}=0

Compare this to the standard 2^{nd} order form

s^{2} + 2 ζ ω_{n} + ω_{n}^{2} = 0

For the two equations to be equal each terms in the two equations must match. Set ζ=0.707, match terms and solve:

2 = 2 ζ ω_{n}=2 (0.707)ω_{n}

and

0.1 k_{p} = ω_{n}^{2}

Solve for kand

0.1 k

ω_{n} = 2/(2 (0.707)) = 1.414

and

0.1 k = ω_{n}^{2} k = 1.414^{2}/.1 = 20.0

and

0.1 k = ω

Alternately if we know the desired close-loop pole locations we can solve for k_{p} directly. The key is that you must know the exact value for the pole locations and those must lie on the root locus. The desired pole locations from the root-locus graph are: s= -0.025±-0.025 i. Therefore the desired closed-loop equation must be

(s + 1 + 1 i)(s + 1 - 1 i)=0

s^{2} + 2 s + 2

The actual closed-loop characteristic equation is s

s^{2} + 2 s + 0.1 k_{p}

matching terms with the closed-loop equation yields s^{2} = s^{2}

2 s = 2 s

2 = 0.1 k_{p} k_{p} = 20

2 s = 2 s

2 = 0.1 k

Use the following simulation to verify the solution. Select a value for k and change the value for θ_{desired}. The settling time and damping ratio for the system should match the calculate settling time (remind me). Experiment with different value for k and verify that you calculated closed-loop poles (T_{settling} and ζ) match the simulated response.

When k_{p} = 20 the system has closed-loop poles at -1 ± 1 i. This corresponds to a settling time of

The damping ratio (ζ) of 0.707 is a response that has a slight overshoot but does not oscillate.

T_{settling} = -4/(real part of pole location)=4 sec

The damping ratio (ζ) of 0.707 is a response that has a slight overshoot but does not oscillate.

In addition to tracking θ_{desired} the system also rejects disturbances to the motor. Click and drag on the wheel. Note how the torque increases to resist the disturbance.

0

k_{p} =

20.0

The final check in the design is to verify that the motor torque requirements don't exceed the physical limits of the system. Starting at 0° set θ_{desired} to 90° (or some value that represents the maximum change in position) and observe the maximum torque required by the control system. If this value exceeds the physical limits of the system you may need to reduce k_{p}, limit the maximum change in the system, or limit how fast the system is allowed to change (this technique is discussed later in this text).